Weakly interacting one-dimensional topological insulators: A bosonization approach

We investigate the topological properties of one-dimensional weakly interacting topological insulators using bosonization. To do that we study the topological edge states that emerge at the edges of a model realized by a strong impurity or at the boundary between topologically distinct phases. In the bosonic model, the edge states are manifested as degenerate bosonic kinks at the boundaries. We first illustrate this idea on the example of the interacting Su-Schrieffer-Heeger (SSH) chain. We compute the localization length of the edge states as the width of an edge soliton that occurs in the SSH model in the presence of a strong impurity. Next, we examine models of two capacitively coupled SSH chains that can be either identical or in distinct topological phases. We find that weak Hubbard interaction reduces the ground-state degeneracy in the topological phase of identical chains. We then prove that, similarly to the noninteracting model, the degeneracy of the edge states in the interacting case is protected by chiral symmetry. We then study topological insulators built from two SSH chains with interchain hopping that represent models of different chiral symmetric universality classes. We demonstrate in bosonic language that the topological index of a weakly coupled model is determined by the type of interchain coupling, invariant under one of two possible chiral symmetry operators. Finally, we show that a general one-dimensional model in a phase with topological index $\nu$ is equivalent at low energies to a theory of at least $\nu$ SSH chains. We illustrate this idea on the example of an SSH model with longer-range hopping.

Figure 1

The quasiclassical bosonic ground state for an interface between topologically distinct phases with $\delta t>0$ and $\delta t<0$ of a noninteracting SSH chain (6). The kinks in the bosonic field $\varphi \left(x\right)$ at the boundaries correspond to the edge states. We identify the phase with $\delta t>0$ with a trivial vacuum, and the phase $\delta t>0$ with a topologically nontrivial phase.

Figure 2

Possible ground-state configurations of the interacting SSH model in the CDW phase (solid lines). The system spontaneously breaks ${\mathbb{Z}}_2$ symmetry in the bulk by choosing one of the minima shown in red or blue. For a given bulk state, there is a unique kink structure at the edges with lowest energy. Such states differ from the topological edge states in noninteracting SSH model that have an edge degeneracy.

Figure 3

The localization length of the edge states in interacting SSH model with nearest-neighbor interactions as a function of the interaction strength $g=a_0{V}_0$, where $V_0$ is the interaction strength in a lattice model. The localization length is measured in the units $v_F/{\mathrm{\Lambda }}_0$.

Figure 4

(a) Profiles of the charge and spin fields for two identical SSH chains in topologically nontrivial phase with $\delta t<0$. (b) Field configurations for fourfold degenerate many-body states. The column “State” illustrates the fermionic occupation for each of the states in real space. Full circle corresponds to the occupied edge state and an empty circle corresponds to an empty state.

Figure 5

Profiles of the charge and spin fields for inequivalent SSH chains with $\delta t<0$. Kinks of bosonic fields at the boundaries have the same magnitude and correspond to the edge states. The ground state at half-filling is twofold degenerate, the two degenerate states are marked by green/red colors. The green state corresponds to an occupied state in a nontrivial chain, the red one corresponds to an empty state.

Figure 6

Coupled chain model of the topological classes (a) with chiral symmetry $C_1$ that describe classes BDI/CII or AIII and (b) with chiral symmetry $C_2$ that describes classes CI, DIII, or AIII. Chiral symmetry $C_1$ allows coupling between $A$ and $B$ atoms only, while $C_2$ allows interchain coupling only between atoms of the same type.

Figure 7

(a) Illustration of the hopping structure in the extended SSH model (36). (b) Phase diagram of the extended SSH model characterized by the winding number $\nu$.

Figure 8

(a) The spectrum of the extended SSH model at the transition line $t^{\prime }=v$ between the phases $\nu =0$ and $\nu =2$. There are two inequivalent Fermi points given by $k_{1,2}=\pm arccos[w/2t^{\prime }]$. (b) Spectrum at the transition line $w/v=t^{\prime }/v+1$ between the phases with $\nu =1$ and $\nu =0,2$. It crosses the Fermi level at $\pm k_F$, where $k_F=\pi /2$.

Figure 9

(a) Gapless part of the spectrum of the BDI model (E3) for generic parameters $t<0,t_z>0,t_x\ne 0$, the difference between the Fermi momenta is given by $\delta k=k_{+}-k_{-}=2arctan(t_z/\sqrt{t^2-t_x^2})$. (b) Gapless part of the spectrum of the BDI model (E3) for the case $t_x=0$, $t<0,t_z>0$. The difference between the Fermi momenta $\delta k^0=k_{+}^0-k_{-}^0=2arctan(t_z/t)$.

Figure 10

Spectrum of BDI model in a single-band limit.